Parametrices and exact paralinearisation of semi-linear boundary problems

TL;DR

This paper develops a parametrix framework for semi-linear boundary problems, enabling regularity analysis through paralinearisation and broad space applicability, including H"older, Sobolev, Besov, and Lizorkin--Triebel spaces.

Contribution

It introduces an exact paralinearisation approach for product-type nonlinearities, extending parametrix methods to semi-linear boundary problems with broad functional space coverage.

Findings

Parametrices provide regularity under weak conditions.

Pseudo-locality improves regularity in subdomains.

Framework applies to diverse boundary problems including von Karman equation.

Abstract

The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearisation. The parametrices give regularity properties under weak conditions; improvements in subdomains result from pseudo-locality of type $1,1$-operators. The framework encompasses a broad class of boundary problems in H\"older and $L_p$-Sobolev spaces (and also Besov and Lizorkin--Triebel spaces). The Besov analyses of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation.